25 problems

Let $\widehat{\mathscr{W}}_{3,2}$ be the module category of the $\mathfrak{sl}_3$ minimal model $\mathsf{A}_2(3,2)$. A rigid braided tensor category is a braided tensor category in…

Let $P$ be the weight lattice, $P^+$ the dominant integral weights, $P_{\ell}\subset P^+$ the weights of level $ell$, and let $W_{ell}$ be the affine Weyl group acting on the weigh…

Let $\sigma^{\ell}$ denote spectral flow by $\ell$, and let $\mathcal{M}$ and $\mathcal{N}$ be $\mathfrak{G}$-modules with fusion product $\mathbin{\times}$. Spectral-flow compatib…

Fusion-algebra conjecture. The fusion algebra, or Verlinde algebra, of $V$ is isomorphic to the group algebra of $G(V)/H(V)$ over the ground field.

Finiteness conjecture. There are only finitely many Woronowicz algebras having a given $(R^+,list)$ invariant.

Let $\Gamma$ be a discrete subgroup of $SU(n)$, and consider the modular invariants and fusion rings of the $\widehat{SU(n)}$ WZW theory for $n>2$, or of related generalizations. H…

The objects are adjacency graphs associated with fusion rules and modular invariants, together with the graph data arising from discrete subgroups and McKay-type constructions. Hig…

Let $\mathcal{R}=\mathcal{R}_0$ be an integral fusion ring, and let $(\mathcal{R}_n)$ be the sequence obtained by iterating the integral extension construction of Proposition, star…

Let $P$ be the weight lattice, $P^+$ the dominant integral weights, $P_{\ell}\subset P^+$ the weights of level $ell$, and let $W_{ell}$ be the affine Weyl group with length functio…

Let $n\geq 2$ and let $k=-h^\vee+\frac{p}{2n-1}$ be an exceptional level. Let $\mathcal{K}(L_{p-h^\vee}(\mathfrak{so}_{2n+1}))$ be the Grothendieck ring of the corresponding affine…

Let $G$ be a simply-connected, connected, compact Lie group. Multiplication by each element $z\in Z(G)$ of the centre induces an invertible element $J_z\in{}^\kappa KK_G(G,G)$, whi…

Fusion-rule conjecture. There exists such a rational lisse vertex algebra whose irreducible modules satisfy

Let $k$ be an even positive integer, and let $\mathfrak{g}$ be a twisted affine Lie algebra not of type $A_{2\ell}^{(2)}$. Let $P_k^+$ be the set of dominant weights at level $k$,…

Let $\mathfrak{g}$ be an affine Lie algebra of type $A_{2\ell}^{(2)}$, and fix an even level $2n$ with $n\in\mathbb{Z}_{>0}$. Let $P_{2n}$ denote the weights at level $2n$, let…

Let $(\widetilde N^{ab}_c,\widetilde s_a;N^{ij}_k,s_i;\mathcal N^{IJ}_K,\mathcal S_I;c)$ denote the specified fusion, spin, modular-extension and central-charge data, subject to th…

Grothendieck–Verlinde fusion ring isomorphism conjecture. The product on the resulting Grothendieck ring may be identified with the Verlinde product under the group isomorphism…

The singlet vertex algebra $mathcal{W}(2,2p-1)$ is considered together with its module category, which is expected to be a braided tensor category, or at least to contain a suitab…

Let $\mathbb{G}$ be a compact quantum group, and suppose that its fusion semiring satisfies the freeness conjecture. Simplicity and unique trace conjecture. The reduced $C^*$-algeb…

Let $\mathbb{G}$ be a free unitary easy quantum group without any non-trivial one-dimensional representation. Let $\mathcal{C}^{\circ,\bullet}$ be its category of noncrossing color…

Let $\mathbb{G}$ be a free unitary easy quantum group. A free fusion semiring is obtained from a set $S$ with an involution and a binary law by taking the free monoid $R(S)$, with…

Let $k$ be an admissible level with $v>1$. For typical modules labelled by $(\ell,\lambda;\Delta_{r,s})$, define the continuum Verlinde coefficients by … The corresponding formulae…

Let $I$ be the node set, let $t_a$ and $h^{\vee}$ be the parameters associated with the Dynkin diagram, and let $W^{(a)}_m$ be the elements of the WZW fusion ring defined from the…

Let $G$ be a simply-connected, compact, connected Lie group, let $k$ be a level, and let $M_k(G)$ be the integer appearing in the twisted nonequivariant groups $^{\kappa}K_*(G)$, w…

For $n\geq 4$, consider free orthogonal quantum groups and their fusion rules. A free fusion ring is a fusion ring in which every element $a\neq 1$ satisfies $a\cdot a^*\neq 1$. Bi…

Let $G$ be a compact, simple, simply connected Lie group of rank $n$, and let $F_k[G]=R[G]/I_k$ be the fusion ring of positive energy representations at level $k$, where $R[G]$ is…